A Computational Model for Metric Spaces

For every metric space X, we define a continuous poset BX such that X is homeomorphic to the set of maximal elements of BX with the relative Scott topology. The poset BX is a dcpo iff X is complete, and ω-continuous iff X is separable. The computational model BX is used to give domain-theoretic proofs of Banach's fixed point theorem and of two classical results of Hutchinson: on a complete metric space, every hyperbolic iterated function system has a unique non-empty compact attractor, and every iterated function system with probabilities has a unique invariant measure with bounded support. We also show that the probabilistic power domain of BX provides an ω-continuous computational model for measure theory on a separable complete metric space X.